Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f g)(4)$$

Step-by-Step Solution

Answer

\( (f \circ g)(4) = 3\)

Step 1: Evaluate the Inner Function

First, evaluate the inner function, \(g(x)\), at \(x = 4\). This gives \(g(4) = 4 - 2\). Calculate the value to get \(g(4) = 2\).

Step 2: Substitute the Result into the Outer Function

Now, substitute the computed value of \(g(4)\) into the outer function \(f(x)\). This gives \(f(g(4)) = f(2)\).

Step 3: Evaluate the Outer Function

Next, evaluate the outer function, \(f(x)\), at \(x = g(4)\) or \(x = 2\). This gives \(f(2) = 2^{2} - 1\). Calculate the value to get \(f(2) = 3\).

Key Concepts

Function Evaluation

Function evaluation is at the core of understanding how mathematical functions work. Essentially, this is about substituting a specific value into a function to find the output. For instance, with the functions given in our exercise, we start by evaluating the 'inner' function first. The inner function in the composition \(f \circ g\) is \(g(x)\). If we wish to evaluate this composition at \(x=4\), we first find out what \(g(4)\) is. This is done by simply replacing \(x\) in the \(g(x)\) formula with 4, yielding \(g(4)=4-2\), which simplifies to \(g(4)=2\).

By understanding function evaluation, you not only learn how to work with single functions but also to manage more complex scenarios such as function composition. This is a critical skill in algebra that will enable you to tackle various mathematical problems with confidence.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging, simplifying, or rewriting algebraic expressions to make them easier to work with or to find their value. After evaluating the inner function in our example, we perform algebraic manipulation by substituting \(g(4)\) into the outer function, \(f(x)\), which leads to the evaluation of \(f(2)\). Substituting \(2\) for \(x\) in \(f(x)\), we then need to compute \(f(2) = 2^2 - 1\). Through these steps, we use essential algebraic skills such as exponentiation and subtraction, eventually finding that \(f(2) = 3\).

Mastering algebraic manipulation allows you to approach and solve a wide range of algebraic problems with ease, as it is integral to turning complex problems into simpler, solvable equations.

Graphing Utility Verification

Graphing utility verification is a method used to confirm the results of algebraic calculations graphically. Today's technology provides tools like graphing calculators and computer software that can plot functions and their compositions.
For example, after calculating \(f(g(4))\) algebraically and getting a result of 3, you could use graphing technology to plot both functions \(f(x)\) and \(g(x)\). Then, specifically check the point where \(x=4\) on \(g(x)\) and follow it though the composition to see if it matches the algebraic result.

With graphing utilities, you can quickly visualize the functions and the accuracy of your algebraic solution, not only ensuring correctness but also deepening your understanding of how the algebraic operations manifest visually on a graph.